Question

Q27


$\mathbf{27}\mathbf{.}Ifa{\cos }^3\theta +3a{\sin }^2\theta \cos \theta =manda{\sin }^3\theta +3a\sin \theta {\cos }^2\theta =n,provethat{\left(m+n\right)}^{2/3}+{\left(m-n\right)}^{2/3}=2a^{2/3}$

Answer 1

Dear student

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{m}={\mathrm{acos}}^3\mathrm{\theta }+3{\mathrm{asin}}^2\mathrm{\theta }\mathrm{cos\theta } \\ \mathrm{n}={\mathrm{asin}}^3\mathrm{\theta }+3\mathrm{asin\theta }{\cos }^2\mathrm{\theta } \\ \mathrm{Consider},\mathrm{LHS} \\ {\left(\mathrm{m}+\mathrm{n}\right)}^{\frac{2}{3}}+{\left(\mathrm{m}-\mathrm{n}\right)}^{\frac{2}{3}} \\ ={\left({\mathrm{acos}}^3\mathrm{\theta }+3{\mathrm{asin}}^2\mathrm{\theta }\mathrm{cos\theta }+{\mathrm{asin}}^3\mathrm{\theta }+3\mathrm{asin\theta }{\cos }^2\mathrm{\theta }\right)}^{\frac{2}{3}}+{\left({\mathrm{acos}}^3\mathrm{\theta }+3{\mathrm{asin}}^2\mathrm{\theta }\mathrm{cos\theta }-{\mathrm{asin}}^3\mathrm{\theta }-3\mathrm{asin\theta }{\cos }^2\mathrm{\theta }\right)}^{\frac{2}{3}} \\ ={\mathrm{a}}^{\frac{2}{3}}{\left[{\cos }^3\mathrm{\theta }+{\sin }^3\mathrm{\theta }+3\mathrm{cos\theta }\mathrm{sin\theta }\left(\mathrm{cos\theta }+\mathrm{sin\theta }\right)\right]}^{\frac{2}{3}}+{\mathrm{a}}^{\frac{2}{3}}{\left[{\cos }^3\mathrm{\theta }-{\sin }^3\mathrm{\theta }-3\mathrm{cos\theta }\mathrm{sin\theta }\left(\mathrm{cos\theta }-\mathrm{sin\theta }\right)\right]}^{\frac{2}{3}} \\ ={\mathrm{a}}^{\frac{2}{3}}{\left[{\left(\mathrm{cos\theta }+\mathrm{sin\theta }\right)}^3\right]}^{\frac{2}{3}}+{\mathrm{a}}^{\frac{2}{3}}{\left[{\left(\mathrm{cos\theta }-\mathrm{sin\theta }\right)}^3\right]}^{\frac{2}{3}} \\ ={\mathrm{a}}^{\frac{2}{3}}\left[{\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }+2\mathrm{sin\theta cos\theta }+{\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }-2\mathrm{sin\theta cos\theta }\right] \\ ={\mathrm{a}}^{\frac{2}{3}}\times 2\times \left({\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }\right) \\ =2{\mathrm{a}}^{\frac{2}{3}} \\ =\mathrm{RHS} \\ \mathrm{Identity}\mathrm{used}: \\ 1){\cos }^2\mathrm{x}+{\sin }^2\mathrm{x}=1 \end{gathered}$$
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